1. Notes by Shufang Wu http://www.sfu.ca/~vswu
Embedded Block Coding with Optimized Truncation - An Image Compression Algorithm
Notes by Shufang Wu

2. Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides )
Optimal Truncation ( 3 slides )
Embedded Block Coding ( 6 slides )
Abstract Quality Layers ( 1 slide )
Conclusion
References
Q & A

3. Overview (1-1)
Abstract Quality Layers
Division of subbands into code-blocks with the same size
Subband samples for 3 wavelet decomposition levels
What is Embedded Block Coding with Optimized Truncation (EBCOT) ?
Highly scalable bit-stream
Independent embedded blocks
Post-compression rate-distortion (PCRD) optimization
Layered bit-stream organization
Input of EBCOT
Subband samples generated using a wavelet transform, like EZW and SPIHT 
Scalability: A key requirement driving the JPEG2000 standardization process.   
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4. Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides )
Optimal Truncation ( 3 slides )
Embedded Block Coding ( 6 slides )
Abstract Quality Layers ( 3 slides )
Conclusion
References
Q & A

5. Highly Scalable Compression (2-1)
A Scalable Bit-stream
A Highly Scalable Bit-stream
Highly Scalable Compression
A Scalable Bit-stream: One that may be partially discarded to obtain an efficient representation or a lower resolution version of the original image at a different bit-rate.
A Highly Scalable Bit-stream: One that may be decompressed in many different ways with different results, depending on what information has been discarded.
Compress Once: Decompress Many Ways
Refers to the generation of a highly scalable bit-stream

6. Highly Scalable Compression (2-2)
A Resolution Scalable Bit-stream
A Distortion Scalable Bit-stream
A Spatial Scalable Bit-stream (random access)
Elements of a distortion & resolution scalable bit-stream
Quality Progression
Resolution Progression
Resolution Scalable: Containing distinct subsets representing successive resolution levels.
Wavelet transform is an important tool in the construction of resolution scalable bit-streams.
Distortion Scalable: Containing distinct subsets successively augmenting the quality (reducing the distortion).
Embedded quantization is central to the construction of distortion scalable bit-streams.
Random Access: Identifying the region within each subband and hence the code-blocks which are required to correctly reconstruct the region of interest. (Not in detail)

7. Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides )
Optimal Truncation ( 3 slides )
Embedded Block Coding ( 6 slides )
Abstract Quality Layers ( 3 slides )
Conclusion
References
Q & A

8. Optimal Truncation (3-1)
Post-Compression Rate-Distortion (PCRD) Optimization
Rate distortion (R-D) optimization
PCRD
Problem Description
Truncate each code-block bit-stream in an optimal way so as to minimize distortion subject to the bit-rate constraint.
The rate-distortion algorithm is applied after all the subband samples have been compressed.
Optimal selection of the truncation points, { ni }, so as to minimize overall distortion, D, subject to an overall length constraint, Lmax .
D =  Dini, Lmax  L =  Lini
Select ni truncated to Lini with distortion Dini for code-block Bi

9. Optimal Truncation (3-2)
Optimal Selection of Truncation Points
Simple Algorithm to Find ni in Each Code-block for Given 
{ ni }, which minimizes (D() + L()) for some  is optimal.(*)
D cannot be reduced without also increasing L and vice-versa.
 s.t. (*) and L() = Lmax, then { ni } must be an optimal solution. Discrete,  in practice the smallest  s.t. L()  Lmax.
Initialization: ni = 0;
Loop: For j = 1,2, , Zi; (Zi is the last truncation point)
Set L = Lij  Li ni and D = Di ni  Dij ;
IF D/L > , THEN ni = j.
Guarantees that:
Dini + Lini  Diz + Liz for all z  j.
The amount by which the length is increased.
The amount by which the distortion is decreased.

10. Optimal Truncation (3-3)
Set of Feasible Truncation Points, Pi , for code-block, Bi
Distortion-rate slope of a truncation point
Property of the slopes
Reduced Algorithm
Convex Hull Interpretation
Distortion
Truncation Points
An enumeration of points in Pi, 0 = hi0 < hi1 <  < hi|Pi-1|.
Feasible Truncation Points
i(hin) = (Di hin-1  Di hin)/(Li hin  Li hin-1),  n  1.
i(hi0) > i(hi1) > i(hi2) >  > i(hi|Pi-1|). (Strictly decreasing.)
ni = max{ hik  Pi | i(hik) >  }.
Length

11. Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides )
Optimal Truncation ( 3 slides )
Embedded Block Coding ( 6 slides )
Abstract Quality Layers ( 3 slides )
Conclusion
References
Q & A

12. Embedded Block Coding (6-1)
Bit-plane coding
Encode first the most significant bits for all samples.
Basics
Classical context adaptive arithmetic coding (18)
Bit-plane coding
Fractional bit-plane coding (Passes)
Sub-blocks with explicit sub-block significance coding
Purpose: Ensure a sufficiently fine embedding.
Assumption: Significant samples tend to be clustered.

13. Embedded Block Coding (6-2)
Quantization
Scalar quantizer (SQ)
Deadzone uniform scalar quantizer
Embedded deadzone quantizers
Significance of A Sample
A state variable, initialized to 0, transitioned to 1 when its first nonzero bit-plane is encoded
A function that maps each element in a subset of the real line to a particular value in that subset. 1 x
0.5
1.5 -1
-0.5
-1.5
SQ with the interval about 0 widened ( twice as large ).
Family of embedded deadzone quantizers
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14. Embedded Block Coding (6-3)
Sub-Block Significance Coding ( 16  16 )
For each bit-plane, first encode information to identify those sub-blocks that contain one or more significant samples
One way to encode is quad-tree coding
Bit-Plane Coding Primitives
4 primitives: Zero coding (ZC), Run-length coding (RLC), Sign coding (SC) and Magnitude refinement (MR)
How to use these primitives ?
In every case, the symbol must be coded using arithmetic coder with 18 probability models
Embedded Quad-tree Structure
Identifying Significance
Constructing Tree
Entire Code-block
IF the sample is not yet significant, ZC or RLC is used;
IF it is significant, SC is used;
ELSE (the sample is already significant) MR is used.
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18 Probability Models: 9 for ZC; 1 for RLC; 5 for SC; 3 for MR.

15. Embedded Block Coding (6-4)
Zero Coding (ZC)  9
To code pth bit of the quantized magnitude M, given that the sample is not yet significant ( M < 2p+1 )
Dependency ( empirical studies )
Run-Length Coding (RLC)  1
To reduce the number of symbols which must be coded
Invoked in ZC when sample and its neighbors are all insignificant
A group of four samples satisfying 4 conditions is encoded as a single symbol to identify whether any is significant
1) Four consecutive samples must all be insignificant;
2) Their neighbors must all be insignificant;
3) They must reside within the same sub-block;
4) The horizontal index of the first sample must be even.
Formation of Significance Coding Contexts
horizontal
diagonal
vertical
Note: Neighbors outside the code-block are insignificant.
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16. Embedded Block Coding (6-5)
Sign Coding (SC)  5 (concerning the signs of 4 neighbors)
Used at most once when found significant in ZC or RLC
Redundancy exploited ( 2 symmetry properties )
Magnitude Refinement (MR)  3
To code pth bit of the quantized magnitude M, given that M  2p+1
Dependency ( empirical studies )
0: Both neighbors are insignificant or
significant with different signs;
1: At least one neighbor is positive;
-1: At least one neighbor is negative.
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Context 0: S = 0 and 4 neighbors are insignificant;
Context 1: S = 0 and at least one neighbor is significant;
Context 2: S = 1. (S is a state variable that transitions from 0 to 1 after MR is first applied to the sample.)

17. Embedded Block Coding (6-6)
Fractional Bit-Planes (4)
For each bit-plane, the coding proceeds in some passes and each sample appears in exactly one of the passes
Forward Significance Propagation Pass ( 1p for bit-plane p )
Skipping over all samples either insignificant or that do not have a preferred neighborhood(PN)
Reverse Significance Propagation Pass ( 2p )
Identical to 1p except two points
Magnitude Refinement Pass ( 3p )
Already significant and no information has been coded
Normalization Pass ( 4p )  all sample not considered before
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Appearance of coding passes and quad-tree codes in each block’s embedded bit-stream
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LL & LH: At least one horizontal neighbor is significant;
HL: At least one vertical neighbor is significant;
HH: At least one diagonal neighbor is significant.
1) Samples are visited in the reverse order;
2) PN is expanded: At least one of 8 neighbors is significant.

18. Agenda Overview ( 1 slide ) Highly Scalable Compression ( 2 slides )
Optimal Truncation ( 3 slides )
Embedded Block Coding ( 6 slides )
Abstract Quality Layers ( 3 slides )
Conclusion
References
Q & A

19. Abstract Quality Layers (1-1)
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Tier 1
Embedded block coding
Operates on block samples
Tier 2
Coding of block contributions to each quality layer
Operates on block summary information
Code-block samples
Compressed image
Embedded block bit-streams
Second Tier of Two-tiered Coding Structure
Abstract Quality Layers
Information Must Be Explicitly Identified
Length of the segment
Number of new coding passes
pimax, most significant bit-plane of code-block, Bi
qi , index of first layer Bi makes nonempty contribution
Substantial inter-block redundancy

20. Conclusion State-of-the-art Compression Performance
Embedded Block Coding
High Scalability (3)
Resolution scalability
Distortion/SNR/quality/rate scalability
Spatial scalability (Random access capability)
Enabling
Use of Post-Compression Rate-Distortion Optimization
Introducing
the Concept of Abstract Quality Layers

21. References
1. D. S. Taubman, “High performance scalable image compression with EBCOT,” IEEE Trans. Image Proc., vol. 9, pp , July 2000
2. D. S. Taubman and M. W. Marcellin, JPEG2000: Image Compression Fundamentals, Standards and Practice, Kluwer Academic Publishers, 2002
3. J. M. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Processing., vol. 41, pp , December 1993
4. A. Said and W. Pearlman, “A new, fast and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, pp , June 1996
5. S. Wu, “Embedded zerotree wavelet: an image coding algorithm”, June 2002